A fundamental goal in constitutive modeling is the ability to predict the mechanical behavior of a material under a generalized loading state. To achieve this goal, rigorous experimentation involving all relevant deformations is necessary to obtain both the form and material constants of a strain-energy density function. For both natural biological tissues and tissue-derived soft biomaterials, there exist many physiological, surgical, and medical device applications where rigorous constitutive models are required. Although able to fit the biaxial data well, phenomenological models cannot be used to determine the underlying mechanisms of tissue mechanical behavior. In particular, the respective roles of the fibers and the matrix and how these may change with growth or chemical treatments are unknown. Structurally based constitutive models avoid ambiguities in material characterization and offer insights into the function, structure, and mechanics of tissue components. In the present work a structural constitutive model for the aortic valve is presented as an example of a structural approach. Ongoing issues in practically applying structural models to other tissues are also addressed.